Question

one study claimed that 65% of teenagers wear a seat belt while driving. a parent thinks the actual proportion is lower than 65%. the parent conducts a simple random sample of 130 teenagers, and finds that 75 of them typically wear a seat belt while driving. use a 0.05 level of significance for the hypothesis test. which of the following represents the alternative hypothesis? a. p ≠ 0.65 b. p < 0.65 c. p = 0.65 d. p ≠ 0.65 e. μ < 0.65 question 17 find the value of the sample proportion p hat. round to two decimal places. answer: question 18 find the critical value. answer: question 19 find the value of the test statistic. round to two decimal places. you should use p = 0.65, and the p hat should also have two decimal places. answer: question 20 find the p - value. you should have four decimal places. answer: question 21 based on either the p - value compared to the level of significance, or by drawing a rejection region, make a conclusion. a. reject the null hypothesis b. fail to reject the null hypothesis

Explanation:

Question 16

The alternative hypothesis for a claim that the proportion is lower than a certain value is \(p < 0.65\).

Question 17

Step1: Calculate sample proportion

The sample proportion \(\hat{p}=\frac{x}{n}\), where \(x = 75\) and \(n=130\).
\(\hat{p}=\frac{75}{130}\approx0.58\)

Question 18

Step1: Find critical value for one - tailed test (\(\alpha = 0.02\))

Using standard normal distribution \(z\) - table, for a left - tailed test with \(\alpha=0.02\), the critical value \(z_{c}=- 2.05\)

Question 19

Step1: Calculate test statistic

The formula for the test statistic \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Given \(p = 0.65\), \(\hat{p}=0.58\), \(n = 130\)
\(z=\frac{0.58 - 0.65}{\sqrt{\frac{0.65\times(1 - 0.65)}{130}}}=\frac{-0.07}{\sqrt{\frac{0.65\times0.35}{130}}}\)
First, calculate \(\sqrt{\frac{0.65\times0.35}{130}}=\sqrt{\frac{0.2275}{130}}\approx\sqrt{0.00175}\approx0.042\)
Then \(z=\frac{- 0.07}{0.042}\approx - 1.67\)

Question 20

Step1: Find p - value

Using standard normal table for \(z=-1.67\), the p - value is \(P(Z < - 1.67)=0.0475\)

Question 21

Step1: Compare p - value and \(\alpha\)

Since \(p - value=0.0475>0.02=\alpha\), we fail to reject the null hypothesis

Answer:

Question 16: b. \(p < 0.65\)
Question 17: \(0.58\)
Question 18: \(-2.05\)
Question 19: \(-1.67\)
Question 20: \(0.0475\)
Question 21: b. Fail to reject the null hypothesis